Pressure loss coefficient

Physical key figure
Surname	Pressure loss coefficient
Formula symbol	${\displaystyle \zeta }$
dimension	dimensionless
definition	${\displaystyle \zeta =2{\frac {\Delta p}{\rho \cdot v^{2}}}}$
${\displaystyle \Delta p}$ Pressure loss ${\displaystyle v}$ mean speed in the reference cross-section ${\displaystyle \rho }$ density
scope of application	Flow through components

The pressure loss coefficient , pressure loss coefficient or resistance coefficient (common symbol ( zeta )) is a dimensionless measure in fluid mechanics for the pressure loss in a component such as a pipeline or valve . This means that the pressure loss coefficient says something about the pressure difference between the inflow and outflow to maintain a certain flow through the component. The pressure loss coefficient always applies to a certain geometric shape and is generally dependent on the Reynolds number and, if applicable, on the surface roughness. ${\displaystyle \zeta }$ ${\displaystyle {\mathit {Re}}}$

The drag coefficient described here for components with a flow corresponds to the pressure coefficient between the inlet and outlet pressure and is the analogue of the drag coefficient (also known as the flow resistance coefficient) for bodies with a flow.

definition

The pressure loss coefficient is defined as follows: ${\displaystyle \zeta }$

${\displaystyle \zeta ={\frac {\Delta p}{{\frac {\rho }{2}}\cdot v^{2}}}}$

Here, the pressure loss in the portion (eg., Valve or elbow) and the average speed in a specific reference cross-section. The specification of the coefficient only makes sense together with the definition of the reference cross-section. ${\displaystyle \Delta p}$ ${\displaystyle v}$

The drag coefficient given for individual components generally relates to the installation of the component in a duct or a pipe section and describes the additional pressure loss that results from the insertion of the component. Resistance coefficients of components connected in series can be added as long as they refer to the same reference cross-section.

The pressure loss calculation of individual resistances can be carried out using the zeta value or using the kv value or using the flow coefficient. These three quantities can be converted into one another. Information on this and special individual resistance coefficients for pipe branches and pipe unions, which are also suitable for creating computer programs, can be found in: ${\displaystyle \alpha }$ ${\displaystyle \zeta }$

Hints

Series connection of flow resistances

Drag coefficients of flow components can only be added if there is no mutual influence. This is usually only guaranteed if the components are spaced sufficiently apart. When several elements are directly coupled, the drag coefficients can increase considerably (example: weather protection grille with silencer). As a rule, an empirical determination of the drag coefficient of the combination is then necessary.

Resistances of freely blowing components

Resistance coefficients are usually determined when the components are installed in ducts or pipes. This can have a significant impact. Example: In the case of silencers, the free duct section acts as a shock diffuser and leads to pressure recovery. This considerably reduces the drag coefficient. There is no pressure recovery with freely blowing out silencers, the drag coefficient can increase up to twice the catalog value.

Resistances of free flow components

When specifying resistance coefficients for components exposed to free flow (e.g. weather protection grilles in facades), misunderstandings often arise. The still ambient air is accelerated to the mean flow velocity with an ideal opening. This reduces the static pressure in the opening. This is not a loss of pressure as the kinetic energy increases by the amount that the pressure energy decreases. The energy of the fluid elements remains constant, there is no pressure loss (i.e. conversion of pressure energy into heat energy), but a theoretically reversible pressure change.

For an ideally rounded inlet ( or ), this change in static pressure is calculated ${\displaystyle c_{v}=1}$ ${\displaystyle \zeta _{E}=0}$

${\displaystyle \Delta p_{S}={\frac {\rho }{2}}\cdot v^{2}}$

The total pressure or the energy content

${\displaystyle p_{t}={\frac {\rho }{2}}\cdot v^{2}+p_{s}}$

remains constant along the streamline (with an ideal lossless inlet). The kinetic energy contained is usually only lost when the air jet emerges into the open at the end of the system. There occurs , based on the average speed in the exit cross-section. That is why diffusers are often used to reduce the speed in the outlet and thus this loss. ${\displaystyle \zeta _{A}=1}$

Real values ​​of can be achieved with aerodynamically cleverly designed inlets , a simple hole brings it to about (in each case based on the cross-section of the connecting pipe section). ${\displaystyle \zeta _{E}=0{,}05}$ ${\displaystyle \zeta _{E}=0{,}6}$

Pressure loss coefficient and flow coefficient c _v

When testing smoke and heat exhaust openings and similar air outlets, it is common to specify flow coefficients ( value). These indicate the ratio of the geometric to the aerodynamically effective opening area. Since the value is not suitable for adding up flow components, it must be converted into a drag coefficient. The following conversions apply: ${\displaystyle c_{v}}$ ${\displaystyle c_{v}}$

a) In the case of closed flows (pipes, channels, etc. or network of the same)

${\displaystyle c_{v}={\sqrt {\frac {1}{1+\zeta }}}\quad \Leftrightarrow \quad \zeta ={\frac {1}{c_{v}^{2}}}-1}$

This conversion is based on the definition that applies to an ideal flow element that does not generate any pressure loss . ${\displaystyle c_{v}=1\rightarrow \zeta =0}$

b) In the case of inflow / outflow openings

${\displaystyle c_{v}={\sqrt {\frac {1}{\zeta }}}\quad \Leftrightarrow \quad \zeta ={\frac {1}{c_{v}^{2}}}}$

This definition takes into account that the dynamic pressure q (dynamic pressure) of the flow on the A or outflow is lost and the system no longer available. In the case of an ideal inflow / outflow element, the result is based on the inflow / outflow cross section. ${\displaystyle c_{v}=1}$ ${\displaystyle \zeta =1}$

application

By knowing the resistance coefficients of all sections, the total pressure drop of a pipeline system or sewer network can be determined. This is important for the design of the conveyor device (e.g. pump or fan ).

literature

A comprehensive description of known resistance coefficients can be found in: IE Idel'chik: Handbook of hydraulic resistance . Begell House

Remarks

1. ↑ also with scaled enlargement / reduction of the component
2. ↑ Bernd Glück: "Hydrodynamic and gas-dynamic pipe flow, pressure losses" . Pressure drop algorithms for programming
3. ↑ Fiedler, E., On the pressure loss calculations, especially from simulation results, Bauphysik 31, Issue 6, 2009. "